Geometry Workbook — Chapter 16: Surface Area and Volume of Solids

SOL G.DF.1 (a, b, c, d) · Companion to Textbook Chapter 16

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 124.

PAGE 1 — Chapter opener

Chapter 16 · Surface Area and Volume of Solids

Standard G.DF.1 (a, b, c, d)

In this chapter you will:

• Identify the two-dimensional cross section produced when a plane cuts a three-dimensional figure
• Find the volume and surface area of rectangular and triangular prisms
• Find the volume and surface area of cylinders and cones, using the Pythagorean Theorem to find a slant height
• Find the volume and surface area of square pyramids and spheres
• Find the volume and surface area of composite solids built from two solids
• Work backward from a given surface area or volume to find an unknown dimension

Words to know: cross section · prism · cylinder · cone · pyramid · sphere · hemisphere · apex · slant height · lateral surface · composite solid

Conventions: a slant height $l$ is always found with the Pythagorean Theorem, never given directly. Answers with $\pi$ stay an exact coefficient, never a rounded decimal. A composite solid's surface area adds each piece's full surface area, then subtracts the shared face TWICE.

PAGE 2 — The whole chapter on one page

One Idea, Five Solids

Every "surface area needs" column that mentions $l$ is a formula the EOC formula sheet cannot finish for you — $l$ always has to be found first, with the Pythagorean Theorem.

PAGE 3 — Cuts parallel to the base

16.1 The Simplest Cut

FIGURE: fig1-cross-sections-parallel-to-the-base.png (full width)

Fill in.

prism → ______________    cylinder → ______________    cone → ______________    sphere → ______________

Which one of these four shrinks as the cut moves toward the top? ______________

PAGE 4 — Cuts through the axis or the apex

16.1 A Different Kind of Cut

FIGURE: fig2-cross-sections-through-the-apex.png (full width)

Fill in.

cylinder through its axis → ______________    cone through its apex → ______________    pyramid through its apex → ______________

PAGE 5 — Guided practice 16.1

Work Through It

1. Name the shape from the prism's cut, and say whether it stays the same size everywhere. ______
2. Name the shape from the cylinder's cut and the cone's cut. What's the one difference? ______
3. Name the shape from the sphere's cut. Why doesn't a sphere need the word "parallel"? ______
4. Name the shape from the cylinder cut through its axis. ______
5. Name the shape from the cone and the pyramid cut through the apex. What do the two cuts have in common? ______
6. What single fact about the cutting plane decides which family of shapes results? ______

PAGE 6 — Name the shape

16.1 Name It

Name the two-dimensional shape produced by each cut.

7. A cylinder is cut by a plane parallel to its base. ______
8. A cylinder is cut by a plane containing its axis. ______
9. A cone is cut by a plane parallel to its base, partway up. ______
10. A cone is cut through its apex, perpendicular to its base. ______
11. A square pyramid is cut through its apex, perpendicular to its base. ______
12. A square pyramid is cut by a plane parallel to its base. ______
13. A rectangular prism is cut by a plane parallel to one of its rectangular faces. ______
14. A triangular prism is cut by a plane parallel to its triangular base. ______

PAGE 7 — Name it, and explain it

16.1 Finish the Set

15. A triangular prism is cut by a plane parallel to one of its rectangular side faces. ______

16. A sphere is cut by a plane through its center. ______

17. A sphere is cut by a plane that misses the center. ______

18. A cylinder is cut by a plane parallel to its base, close to one end. ______

19. Application. A carpenter saws through a triangular-prism doorstop, parallel to its triangular ends. Name the exposed shape. ______

20. Error analysis. A student claims a cone's parallel cross section is always the same size as the base. Explain the error. ______

21. Reasoning. Why is a sphere's cross section always a circle, while a cylinder's or cone's is only a circle for cuts parallel to the base? ______

Exit ticket 16.1

22. A plane cuts a cylinder parallel to its base. Name the shape, and say whether its size depends on where the cut is made. ______
23. A plane cuts a square pyramid through its apex. Name the shape. ______

PAGE 8 — A rectangular prism

16.2 Three Pairs of Matching Faces

FIGURE: fig3-rectangular-prism.png (full width)

$$V = lwh \qquad SA = 2(lw+lh+wh)$$

Volume multiplies the three edges once. Surface area adds all six faces — three matching pairs, so each product is doubled.

PAGE 9 — A triangular prism

16.2 Area Times Depth

FIGURE: fig4-triangular-prism.png (full width)

$$V = (\text{area})(\text{depth}) \qquad SA = 2(\text{area}) + (\text{perimeter})(\text{depth})$$

The triangle's equal side is not given — it's the hypotenuse of the right triangle formed by half the base and the height.

PAGE 10 — Guided practice 16.2

Work Through It

24. Identify $l$, $w$, $h$ on the rectangular prism, and give $V$. ______
25. Give its $SA$, and explain why $lw$, $lh$, $wh$ are each doubled. ______
26. Identify the base, height, and depth on the triangular prism, and give the triangle's area. ______
27. Explain how the equal side of $5$ was found, and give the perimeter. ______
28. Give the triangular prism's volume. ______
29. Give its surface area, and name the "two matching ends." ______

PAGE 11 — Rectangular prisms

16.2 Find V and SA

Find the volume and surface area.

30. $l=2$, $w=3$, $h=4$ ______
31. $l=5$, $w=4$, $h=3$ ______
32. $l=7$, $w=3$, $h=2$ ______
33. $l=9$, $w=5$, $h=2$ ______
34. a cube with edge length $6$ ______
35. $l=8$, $w=4$, $h=5$ ______
36. $l=10$, $w=2$, $h=2$ ______
37. $l=12$, $w=5$, $h=3$ ______

PAGE 12 — Triangular prisms

16.2 Find the Leg First

Find the equal side, then the volume and surface area.

38. base $12$, height $8$, depth $5$ ______

39. base $10$, height $12$, depth $7$ ______

40. base $16$, height $15$, depth $6$ ______

41. base $18$, height $12$, depth $4$ ______

42. Application. A cedar chest is $4$ ft by $2$ ft by $3$ ft. Find its volume and surface area. ______

43. Error analysis. A student computes $SA=2lwh=2(60)=120$ for $l=5,w=4,h=3$. Find the error and the correct $SA$. ______

44. Reasoning. Why are the two prisms' $SA$ formulas really the same idea? ______

Exit ticket 16.2

45. $l=6$, $w=3$, $h=2$. Find $V$ and $SA$. ______
46. Triangular prism: base $12$, height $8$, depth $3$. Find $V$ and $SA$. ______

PAGE 13 — A cylinder

16.3 Two Circles and a Rolled-Up Rectangle

FIGURE: fig5-cylinder.png (full width)

$$V = \pi r^2h \qquad SA = 2\pi r^2+2\pi rh$$

PAGE 14 — A cone

16.3 One Circle, One Point

FIGURE: fig6-cone.png (full width)

$$V = \frac{1}{3}\pi r^2h \qquad SA=\pi r^2+\pi rl \qquad l^2=r^2+h^2$$

A cone's volume is always exactly $\frac{1}{3}$ of a cylinder's with the same $r$ and $h$.

PAGE 15 — Guided practice 16.3

Work Through It

47. Identify $r$ and $h$ on the cylinder, and give $V$. ______
48. Give its $SA$. What does $2\pi rh$ represent if the side were unrolled flat? ______
49. Identify $r$ and $h$ on the cone, and find $l$. ______
50. Give the cone's volume. ______
51. Give its $SA$. Why only one $\pi r^2$ term, not two? ______
52. Both formulas start with $\pi r^2$. What does that term represent in each solid? ______

PAGE 16 — Cylinders

16.3 Find V and SA

Find the volume and surface area.

53. $r=4$, $h=10$ ______
54. $r=5$, $h=6$ ______
55. $r=2$, $h=9$ ______
56. $r=7$, $h=10$ ______
57. $r=3$, $h=12$ ______

PAGE 17 — Cones

16.3 Find l First

Find the slant height, then the volume and surface area.

58. $r=6$, $h=8$ ______

59. $r=5$, $h=12$ ______

60. $r=8$, $h=15$ ______

61. $r=9$, $h=12$ ______

62. $r=7$, $h=24$ ______

63. Application. A rain barrel: $r=4$ ft, $h=9$ ft. Find $V$ and $SA$. ______

64. Error analysis. A student uses $SA=\pi r^2+\pi rh$ for $r=5,h=12$, getting $85\pi$. Find $l$ and the correct $SA$. ______

65. Reasoning. Why is a cone's volume always $\frac{1}{3}$ of a same-$r$-and-$h$ cylinder's? ______

Exit ticket 16.3

66. $r=6$, $h=5$ (cylinder). Find $V$ and $SA$. ______
67. $r=12$, $h=16$ (cone). Find $l$, then $V$ and $SA$. ______

PAGE 18 — A square pyramid

16.4 The Apex-to-Edge-Midpoint Triangle

FIGURE: fig7-square-pyramid.png (full width)

$$V=\frac{1}{3}b^2h \qquad SA=b^2+2bl \qquad l^2=\left(\frac{b}{2}\right)^2+h^2$$

$l$ runs to the MIDDLE of a base edge, not to a corner.

PAGE 19 — A sphere

16.4 One Measurement Does It All

FIGURE: fig8-sphere.png (full width)

$$V=\frac{4}{3}\pi r^3 \qquad SA=4\pi r^2$$

PAGE 20 — All five, side by side

16.4 The Formula Board

FIGURE: fig9-formula-board.png (full width)

The formula sheet supplies every line here — except $l$, which you always find yourself.

PAGE 21 — Guided practice 16.4

Work Through It

68. Identify $\frac{b}{2}$ and $h$ on the pyramid, and find $l$. ______
69. Give the pyramid's volume. ______
70. Give its $SA$. Where does $2bl$ come from? ______
71. Identify $r$ on the sphere, and give $V$. ______
72. Give the sphere's $SA$. Why only one measurement needed? ______
73. On the formula board, which two formulas need a value it doesn't supply? ______

PAGE 22 — Square pyramids

16.4 Find l First

Find the slant height, then the volume and surface area.

74. $b=6$, $h=4$ ______
75. $b=12$, $h=8$ ______
76. $b=10$, $h=12$ ______
77. $b=16$, $h=15$ ______
78. $b=18$, $h=12$ ______

PAGE 23 — Spheres

16.4 Find V and SA

Find the volume and surface area.

79. $r=3$ ______

80. $r=9$ ______

81. $r=12$ ______

82. $r=15$ ______

83. Application. A pyramid monument: square base $40$ ft, height $15$ ft. Find $V$, and the four triangular faces' area. ______

84. Error analysis. A student computes $V=b^2h=1200$ for $b=10,h=12$, forgetting $\frac{1}{3}$. Find the correct $V$. ______

85. Reasoning. A sphere with $r=3$ has $V=SA=36\pi$. Why does $r=3$ do this, and why doesn't it mean $V$ and $SA$ are the same kind of quantity? ______

Exit ticket 16.4

86. $b=14$, $h=24$ (pyramid). Find $l$, then $V$ and $SA$. ______
87. $r=6$ (sphere). Find $V$ and $SA$. ______

PAGE 24 — A silo

16.5 Cylinder and Hemisphere

FIGURE: fig10-silo.png (full width)

Add each piece's full $SA$, then subtract the shared circle TWICE.

PAGE 25 — A house

16.5 Box and Triangular-Prism Roof

FIGURE: fig11-house.png (full width)

$$SA = (\text{box full } SA)+(\text{roof full } SA)-2(\text{shared rectangle})$$

PAGE 26 — An ice-cream cone

16.5 Cone and Hemisphere

FIGURE: fig12-ice-cream-cone.png (full width)

Neither piece keeps a flat circular face here — their one flat face is the SAME shared circle.

PAGE 27 — Guided practice 16.5

Work Through It

88. Identify the silo's cylinder $r$, $h$, and hemisphere radius. ______
89. Give the silo's total volume. Why is the hemisphere's own flat circle left out of $SA$? ______
90. Identify the house's box and roof dimensions, and the shared face's area. ______
91. Give the house's total $V$ and $SA$. Why subtract the shared face twice? ______
92. Identify the ice-cream cone's shared radius. Why does neither piece keep a flat face? ______
93. Give the ice-cream cone's total $V$ and $SA$. ______

PAGE 28 — Build the composite

16.5 Find Total V and SA

Find the total volume and total surface area.

94. Cylinder ($r=4,h=10$) + cone ($r=4,h=3$) — a pencil. ______

95. Box ($6\times6\times5$) + square pyramid ($b=6,h=4$) — a tent. ______

96. Cylinder ($r=6,h=10$) + hemisphere ($r=6$) — a larger silo. ______

97. Cone ($r=6,h=8$) + hemisphere ($r=6$) — a larger ice-cream cone. ______

98. Cylinder ($r=6,h=10$) + hemispheres ($r=6$) on BOTH ends — a capsule. ______

99. Error analysis. A student adds the silo's cylinder full $SA$ ($66\pi$) to the hemisphere's full $SA$ ($27\pi$), getting $93\pi$. Which circle was double-counted? Give the correct $SA$. ______

100. Reasoning. Why does "add full areas, subtract the shared face twice" always match counting only the exposed faces directly? ______

Exit ticket 16.5

101. For the silo, name the circle subtracted twice, and give its area. ______
102. Cone ($r=3,h=4$) + hemisphere ($r=3$). Find total $V$ and $SA$, and list every included surface. ______

PAGE 29 — Two ways to work backward

16.6 Same Formula, Different Letter

FIGURE: fig13-working-backward.png (full width)

$$100\pi=\pi(5)^2h \rightarrow h=4$$

$$96=6^2+2(6)l \rightarrow l=5 \rightarrow h^2=5^2-3^2=16 \rightarrow h=4$$

Substitute everything known, then undo the remaining arithmetic one step at a time. Surface area often takes an extra Pythagorean Theorem step that volume never needs.

PAGE 30 — Guided practice 16.6

Work Through It

103. Identify what's given and unknown in the cylinder problem. ______
104. Show the substitution and solve for $h$. ______
105. Identify what's given and unknown in the pyramid problem. ______
106. Solve for $l$ first. ______
107. Use $l$ to find $h$. ______
108. Compare the two problems: same strategy, different number of steps — explain. ______

PAGE 31 — Solve for the missing dimension

16.6 Find It

109. Rectangular prism: $V=150$, $l=10$, $w=5$. Find $h$. ______
110. Rectangular prism: $SA=94$, $l=5$, $w=4$. Find $h$. ______
111. Cylinder: $V=98\pi$, $r=7$. Find $h$. ______
112. Cylinder: $SA=136\pi$, $r=4$. Find $h$. ______
113. Cone: $V=48\pi$, $r=6$. Find $h$. ______
114. Cone: $SA=90\pi$, $r=5$. Find $l$, then $h$. ______

PAGE 32 — Finish the set

16.6 Find It

115. Sphere: $V=972\pi$. Find $r$. ______

116. Sphere: $SA=400\pi$. Find $r$. ______

117. Square pyramid: $V=400$, $b=10$. Find $h$, then $l$. ______

118. Triangular prism: $V=180$, base $6$, height $4$. Find the depth. ______

119. Triangular prism: $SA=152$, base $6$, height $4$. Find the depth. ______

120. Application. A silo holds $250\pi$ ft³, $r=5$ ft. Find its height. ______

121. Error analysis. For $V=200\pi$, $r=5$, a student writes $200\pi=\pi(5)h$ and gets $h=40$. Find the mistake and the correct $h$. ______

122. Reasoning. Why is working backward from surface area often longer than from volume, for a cone or pyramid? ______

Exit ticket 16.6

123. Cylinder: $V=48\pi$, $r=4$. Find $h$. ______
124. Cone: $SA=216\pi$, $r=9$. Find $l$, then $h$. ______

PAGE 33 — Blank solids

Your Turn

FIGURE: fig14-blank-solid-frames.png (full width)

For every problem in this chapter:

• Label the dimensions you'd need before you try to compute anything.
• For a cone or a pyramid, find $l$ with the Pythagorean Theorem before using it in $SA$.
• For a composite, decide which faces are shared and subtract them TWICE from the total $SA$ — never from $V$.
• Keep $\pi$ exact; keep radicals in simplest form.

PAGE 34 — Chapter review

Chapter 16 Review

Review 1 (G.DF.1a). A rectangular prism, a cylinder, a cone, and a sphere sit on a table.

• A plane cuts the prism parallel to a side face. Name the shape.
• A plane cuts the cylinder through its axis. Name the shape.
• A plane cuts the cone parallel to its base, one-third of the way up. Name the shape, and say if it's the same size as the base.
• A plane cuts the sphere anywhere at all. Name the shape, and explain why the solid guarantees that answer.

Review 2 (G.DF.1 b, c). A shed is a rectangular prism $8$ ft by $6$ ft by $7$ ft, topped with a triangular-prism roof (triangular base $6$ ft, height $4$ ft, running the full $8$-ft length).

• Find the box's volume, the roof's volume, and the total volume.
• Find each part's full surface area, the shared face's area, and the total surface area.
• Explain why the shared face is subtracted twice, not once.

Review 3 (G.DF.1d). A cone has surface area $90\pi$ and radius $5$.

• Find the slant height $l$.
• Find the height $h$.
• Find the volume using your $h$.
• Confirm your answer using $V=\frac{1}{3}\pi r^2h$ directly.